Numerical solution of Y” = F(X,Y) with particular reference to the radical schrödinger eqution

Abstract

Many theoretical treatments of quantum-mechanical scattering processes require the numerical solution of a set of second order ordinary differential equations of special form (with first derivative absent). The methods used to solve such sets of equations are generally based on step-by-step methods for solving a single second order differential equation over a fixed mesh. For example Chandra (1973) has published a computer program which uses de Vogelaere's method to solve the differential equations arising in a close-coupling formulation of quantum mechanical scattering problems. Chandra's program makes no attempt to monitor the local truncation error and leaves the choice of steplength strategy entirely to the user. Our aim is to improve on existing implementations of de Vogelaere's method for a single second order equation by incorporating a method of truncation error estimation and an automatic mesh-selection facility. Estimates of the truncation error in de Vogelaere's method are established together with an upper bound for the local truncation error; the interval of absolute stability is found to be [-2,0] and it is shown that the global truncation error is of order h(^4) where h is the steplength. In addition the characteristics of a method due to Raptis and Allison are investigated. A numerical comparison of computer programs which incorporate the methods of de Vogelaere, Numerov, Raptis and Allison and Adams-Bashforth Adams-Moulton, with an automatic error control is performed to determine which program gives the most reliable and efficient solution of the single channel radial Schrödinger equation. A modification of Chandra's program is provided which performs the numerical integration of a set of coupled second order homogeneous differential equations using de Vogelaere's method with an automatic error control.